NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.2% → 98.7%

Time: 7.1s

Alternatives: 17

Speedup: 1.2×

• 37.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.0578645557035018e+103
  2. eps = 4.757752329810648e-129

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (+ (/ 1.0 (exp (* x (- 1.0 eps)))) (exp (- (fma x eps x)))) 0.5))

C

double code(double x, double eps) {
	return ((1.0 / exp((x * (1.0 - eps)))) + exp(-fma(x, eps, x))) * 0.5;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(1.0 / exp(Float64(x * Float64(1.0 - eps)))) + exp(Float64(-fma(x, eps, x)))) * 0.5)
end

Wolfram

code[x_, eps_] := N[(N[(N[(1.0 / N[Exp[N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 74.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
4. Step-by-step derivation

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

   \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5 \]

8. Final simplification 99.2%

   \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5 \]
9. Add Preprocessing

Alternative 2: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+208}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \frac{1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.6e+208)
   (* (+ (exp (* x eps)) (exp (- (fma x eps x)))) 0.5)
   (if (<= x 1.22e+247)
     (/ (- (- (pow eps -1.0) -1.0) (* (- (/ 1.0 eps) 1.0) 1.0)) 2.0)
     (/
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
       (* (- (/ -1.0 eps) -1.0) (/ 1.0 (fma (- eps -1.0) x 1.0))))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.6e+208) {
		tmp = (exp((x * eps)) + exp(-fma(x, eps, x))) * 0.5;
	} else if (x <= 1.22e+247) {
		tmp = ((pow(eps, -1.0) - -1.0) - (((1.0 / eps) - 1.0) * 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp(((-1.0 + eps) * x))) + (((-1.0 / eps) - -1.0) * (1.0 / fma((eps - -1.0), x, 1.0)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.6e+208)
		tmp = Float64(Float64(exp(Float64(x * eps)) + exp(Float64(-fma(x, eps, x)))) * 0.5);
	elseif (x <= 1.22e+247)
		tmp = Float64(Float64(Float64((eps ^ -1.0) - -1.0) - Float64(Float64(Float64(1.0 / eps) - 1.0) * 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(Float64(-1.0 + eps) * x))) + Float64(Float64(Float64(-1.0 / eps) - -1.0) * Float64(1.0 / fma(Float64(eps - -1.0), x, 1.0)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.6e+208], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.22e+247], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(1.0 / N[(N[(eps - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6e208

   1. Initial program 71.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \left(e^{\varepsilon \cdot x} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.8%

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5 \]

   if 2.6e208 < x < 1.22000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{1}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{1}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left({\varepsilon}^{-1} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2} \]

   if 1.22000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\color{blue}{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      5. exp-neg N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{\color{blue}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{\left(\varepsilon + 1\right)} \cdot x}}}{2} \]

      9. distribute-rgt1-in N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{x + \varepsilon \cdot x}}}}{2} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{\varepsilon \cdot x + x}}}}{2} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{x \cdot \varepsilon} + x}}}{2} \]

      12. lower-fma.f64 100.0

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{\color{blue}{1 + x \cdot \left(1 + \varepsilon\right)}}}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(\varepsilon + 1, x, 1\right)}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+208}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \frac{1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (+ (exp (* x (+ -1.0 eps))) (exp (- (fma x eps x)))) 0.5))

C

double code(double x, double eps) {
	return (exp((x * (-1.0 + eps))) + exp(-fma(x, eps, x))) * 0.5;
}

Julia

function code(x, eps)
	return Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) + exp(Float64(-fma(x, eps, x)))) * 0.5)
end

Wolfram

code[x_, eps_] := N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 74.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
4. Step-by-step derivation

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

6. Final simplification 99.2%

   \[\leadsto \left(e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 64.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+208}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \frac{1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-271)
   (* (+ (/ 1.0 (fma x (- 1.0 eps) 1.0)) (exp (- (* x eps)))) 0.5)
   (if (<= x 2.6e+208)
     (* (- (exp (* x eps)) -1.0) 0.5)
     (if (<= x 1.22e+247)
       (/ (- (- (pow eps -1.0) -1.0) (* (- (/ 1.0 eps) 1.0) 1.0)) 2.0)
       (/
        (+
         (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
         (* (- (/ -1.0 eps) -1.0) (/ 1.0 (fma (- eps -1.0) x 1.0))))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-271) {
		tmp = ((1.0 / fma(x, (1.0 - eps), 1.0)) + exp(-(x * eps))) * 0.5;
	} else if (x <= 2.6e+208) {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	} else if (x <= 1.22e+247) {
		tmp = ((pow(eps, -1.0) - -1.0) - (((1.0 / eps) - 1.0) * 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp(((-1.0 + eps) * x))) + (((-1.0 / eps) - -1.0) * (1.0 / fma((eps - -1.0), x, 1.0)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-271)
		tmp = Float64(Float64(Float64(1.0 / fma(x, Float64(1.0 - eps), 1.0)) + exp(Float64(-Float64(x * eps)))) * 0.5);
	elseif (x <= 2.6e+208)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	elseif (x <= 1.22e+247)
		tmp = Float64(Float64(Float64((eps ^ -1.0) - -1.0) - Float64(Float64(Float64(1.0 / eps) - 1.0) * 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(Float64(-1.0 + eps) * x))) + Float64(Float64(Float64(-1.0 / eps) - -1.0) * Float64(1.0 / fma(Float64(eps - -1.0), x, 1.0)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-271], N[(N[(N[(1.0 / N[(x * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.6e+208], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.22e+247], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(1.0 / N[(N[(eps - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.99999999999999985e-271

   1. Initial program 69.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5 \]

   8. Taylor expanded in eps around inf

      \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\varepsilon \cdot x}\right)\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5 \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\frac{1}{1 + x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5 \]

   if -3.99999999999999985e-271 < x < 2.6e208

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.9%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

      \[\leadsto \left(e^{\varepsilon \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.3%

       \[\leadsto \left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5 \]

   if 2.6e208 < x < 1.22000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{1}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{1}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left({\varepsilon}^{-1} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2} \]

   if 1.22000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\mathsf{neg}\left(\color{blue}{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      5. exp-neg N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{\color{blue}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{\left(\varepsilon + 1\right)} \cdot x}}}{2} \]

      9. distribute-rgt1-in N/A

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{x + \varepsilon \cdot x}}}}{2} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{\varepsilon \cdot x + x}}}}{2} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{x \cdot \varepsilon} + x}}}{2} \]

      12. lower-fma.f64 100.0

          \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\color{blue}{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{\color{blue}{1 + x \cdot \left(1 + \varepsilon\right)}}}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(\varepsilon + 1, x, 1\right)}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+208}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \frac{1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+208}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-271)
   (* (+ (/ 1.0 (fma x (- 1.0 eps) 1.0)) (exp (- (* x eps)))) 0.5)
   (if (<= x 2.6e+208)
     (* (- (exp (* x eps)) -1.0) 0.5)
     (if (<= x 1.22e+247)
       (/ (- (- (pow eps -1.0) -1.0) (* (- (/ 1.0 eps) 1.0) 1.0)) 2.0)
       (* (- (exp (* x (+ -1.0 eps))) -1.0) 0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-271) {
		tmp = ((1.0 / fma(x, (1.0 - eps), 1.0)) + exp(-(x * eps))) * 0.5;
	} else if (x <= 2.6e+208) {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	} else if (x <= 1.22e+247) {
		tmp = ((pow(eps, -1.0) - -1.0) - (((1.0 / eps) - 1.0) * 1.0)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps))) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-271)
		tmp = Float64(Float64(Float64(1.0 / fma(x, Float64(1.0 - eps), 1.0)) + exp(Float64(-Float64(x * eps)))) * 0.5);
	elseif (x <= 2.6e+208)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	elseif (x <= 1.22e+247)
		tmp = Float64(Float64(Float64((eps ^ -1.0) - -1.0) - Float64(Float64(Float64(1.0 / eps) - 1.0) * 1.0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-271], N[(N[(N[(1.0 / N[(x * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.6e+208], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.22e+247], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.99999999999999985e-271

   1. Initial program 69.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5 \]

   8. Taylor expanded in eps around inf

      \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\varepsilon \cdot x}\right)\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5 \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\frac{1}{1 + x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5 \]

   if -3.99999999999999985e-271 < x < 2.6e208

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.9%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

      \[\leadsto \left(e^{\varepsilon \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.3%

       \[\leadsto \left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5 \]

   if 2.6e208 < x < 1.22000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{1}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{1}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left({\varepsilon}^{-1} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2} \]

   if 1.22000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+208}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-271)
   (* (+ (/ 1.0 (fma x (- 1.0 eps) 1.0)) (exp (- (* x eps)))) 0.5)
   (* (- (exp (* x eps)) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-271) {
		tmp = ((1.0 / fma(x, (1.0 - eps), 1.0)) + exp(-(x * eps))) * 0.5;
	} else {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-271)
		tmp = Float64(Float64(Float64(1.0 / fma(x, Float64(1.0 - eps), 1.0)) + exp(Float64(-Float64(x * eps)))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-271], N[(N[(N[(1.0 / N[(x * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999985e-271

   1. Initial program 69.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.2%

      \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5 \]

   8. Taylor expanded in eps around inf

      \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-\varepsilon \cdot x}\right)\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5 \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\frac{1}{1 + x \cdot \left(1 - \varepsilon\right)} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5 \]

   if -3.99999999999999985e-271 < x

   1. Initial program 77.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.2%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

      \[\leadsto \left(e^{\varepsilon \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.4%

       \[\leadsto \left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(\frac{1}{\mathsf{fma}\left(x, 1 - \varepsilon, 1\right)} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\varepsilon - 1\right) \cdot x\\ t_1 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;x \leq -1920000:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+149}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{t\_0 \cdot t\_0 - 1}{t\_0 - 1} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot t\_0 + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (- eps 1.0) x)) (t_1 (+ 1.0 (/ 1.0 eps))))
   (if (<= x -1920000.0)
     (* (- (exp (- x)) -1.0) 0.5)
     (if (<= x -2.9e-246)
       (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
       (if (<= x 9.5e-299)
         (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
         (if (<= x 9e+23)
           (*
            (fma
             (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
             x
             2.0)
            0.5)
           (if (<= x 6.9e+149)
             (/
              (-
               (* t_1 (/ (- (* t_0 t_0) 1.0) (- t_0 1.0)))
               (* (- (/ 1.0 eps) 1.0) (fma -1.0 (fma x eps x) 1.0)))
              2.0)
             (/
              (+ (* t_1 t_0) (* (- (/ -1.0 eps) -1.0) (fma -1.0 x 1.0)))
              2.0))))))))

C

double code(double x, double eps) {
	double t_0 = (eps - 1.0) * x;
	double t_1 = 1.0 + (1.0 / eps);
	double tmp;
	if (x <= -1920000.0) {
		tmp = (exp(-x) - -1.0) * 0.5;
	} else if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 9.5e-299) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= 9e+23) {
		tmp = fma(fma(-1.0, (eps - -1.0), ((-1.0 + (eps * eps)) / (eps - -1.0))), x, 2.0) * 0.5;
	} else if (x <= 6.9e+149) {
		tmp = ((t_1 * (((t_0 * t_0) - 1.0) / (t_0 - 1.0))) - (((1.0 / eps) - 1.0) * fma(-1.0, fma(x, eps, x), 1.0))) / 2.0;
	} else {
		tmp = ((t_1 * t_0) + (((-1.0 / eps) - -1.0) * fma(-1.0, x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps - 1.0) * x)
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (x <= -1920000.0)
		tmp = Float64(Float64(exp(Float64(-x)) - -1.0) * 0.5);
	elseif (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 9.5e-299)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= 9e+23)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(eps - -1.0))), x, 2.0) * 0.5);
	elseif (x <= 6.9e+149)
		tmp = Float64(Float64(Float64(t_1 * Float64(Float64(Float64(t_0 * t_0) - 1.0) / Float64(t_0 - 1.0))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * fma(-1.0, fma(x, eps, x), 1.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_1 * t_0) + Float64(Float64(Float64(-1.0 / eps) - -1.0) * fma(-1.0, x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1920000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-299], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9e+23], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.9e+149], N[(N[(N[(t$95$1 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[(-1.0 * N[(x * eps + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.92e6

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(e^{-1 \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(e^{-x} - -1\right) \cdot 0.5 \]

   if -1.92e6 < x < -2.9e-246

   1. Initial program 56.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x < 9.5000000000000001e-299

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5 \]

   if 9.5000000000000001e-299 < x < 8.99999999999999958e23

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 68.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + 1}\right), x, 2\right) \cdot 0.5 \]

   if 8.99999999999999958e23 < x < 6.9000000000000004e149

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 29.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.0%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{\left(\left(\varepsilon - 1\right) \cdot x\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - 1}{\color{blue}{\left(\varepsilon - 1\right) \cdot x - 1}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   if 6.9000000000000004e149 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 17.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(x \cdot \color{blue}{\left(\varepsilon - 1\right)}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot \color{blue}{x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   12. Taylor expanded in eps around 0

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \color{blue}{-1 \cdot x}\right)}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 38.0%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \color{blue}{x}, 1\right)}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1920000:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+149}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{\left(\left(\varepsilon - 1\right) \cdot x\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - 1}{\left(\varepsilon - 1\right) \cdot x - 1} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1920000:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1920000.0)
   (* (- (exp (- x)) -1.0) 0.5)
   (if (<= x -2.9e-246)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
     (* (- (exp (* x eps)) -1.0) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1920000.0) {
		tmp = (exp(-x) - -1.0) * 0.5;
	} else if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1920000.0)
		tmp = Float64(Float64(exp(Float64(-x)) - -1.0) * 0.5);
	elseif (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1920000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.92e6

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(e^{-1 \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(e^{-x} - -1\right) \cdot 0.5 \]

   if -1.92e6 < x < -2.9e-246

   1. Initial program 56.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x

   1. Initial program 77.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.8%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

      \[\leadsto \left(e^{\varepsilon \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.0%

       \[\leadsto \left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-271)
   (* (+ 1.0 (exp (- (fma x eps x)))) 0.5)
   (* (- (exp (* x eps)) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-271) {
		tmp = (1.0 + exp(-fma(x, eps, x))) * 0.5;
	} else {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-271)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps, x)))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-271], N[(N[(1.0 + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999985e-271

   1. Initial program 69.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.1%

      \[\leadsto \left(1 - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5 \]

   if -3.99999999999999985e-271 < x

   1. Initial program 77.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.2%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

      \[\leadsto \left(e^{\varepsilon \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.4%

       \[\leadsto \left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1920000:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1920000.0)
   (* (- (exp (- x)) -1.0) 0.5)
   (if (<= x -2.9e-246)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
     (if (<= x 9.5e-299)
       (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
       (if (<= x 10500000000.0)
         (*
          (fma
           (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
           x
           2.0)
          0.5)
         (/
          (-
           (* (+ 1.0 (/ 1.0 eps)) (fma (- eps 1.0) x 1.0))
           (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
          2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1920000.0) {
		tmp = (exp(-x) - -1.0) * 0.5;
	} else if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 9.5e-299) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= 10500000000.0) {
		tmp = fma(fma(-1.0, (eps - -1.0), ((-1.0 + (eps * eps)) / (eps - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * fma((eps - 1.0), x, 1.0)) - (((1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1920000.0)
		tmp = Float64(Float64(exp(Float64(-x)) - -1.0) * 0.5);
	elseif (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 9.5e-299)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= 10500000000.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(eps - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64(Float64(1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1920000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-299], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 10500000000.0], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.92e6

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(e^{-1 \cdot x} - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \left(e^{-x} - -1\right) \cdot 0.5 \]

   if -1.92e6 < x < -2.9e-246

   1. Initial program 56.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x < 9.5000000000000001e-299

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5 \]

   if 9.5000000000000001e-299 < x < 1.05e10

   1. Initial program 58.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + 1}\right), x, 2\right) \cdot 0.5 \]

   if 1.05e10 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 24.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.4%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1920000:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.9e-246)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
   (if (<= x 9.5e-299)
     (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
     (if (<= x 10500000000.0)
       (*
        (fma
         (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
         x
         2.0)
        0.5)
       (/
        (-
         (* (+ 1.0 (/ 1.0 eps)) (fma (- eps 1.0) x 1.0))
         (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 9.5e-299) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= 10500000000.0) {
		tmp = fma(fma(-1.0, (eps - -1.0), ((-1.0 + (eps * eps)) / (eps - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * fma((eps - 1.0), x, 1.0)) - (((1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 9.5e-299)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= 10500000000.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(eps - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64(Float64(1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-299], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 10500000000.0], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.9e-246

   1. Initial program 70.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x < 9.5000000000000001e-299

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5 \]

   if 9.5000000000000001e-299 < x < 1.05e10

   1. Initial program 58.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + 1}\right), x, 2\right) \cdot 0.5 \]

   if 1.05e10 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 24.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.4%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.9e-246)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
   (if (<= x 9.5e-299)
     (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
     (if (<= x 5.7e+26)
       (*
        (fma
         (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
         x
         2.0)
        0.5)
       (/
        (+
         (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
         (* (- (/ -1.0 eps) -1.0) (fma -1.0 x 1.0)))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 9.5e-299) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= 5.7e+26) {
		tmp = fma(fma(-1.0, (eps - -1.0), ((-1.0 + (eps * eps)) / (eps - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * ((eps - 1.0) * x)) + (((-1.0 / eps) - -1.0) * fma(-1.0, x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 9.5e-299)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= 5.7e+26)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(eps - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(eps - 1.0) * x)) + Float64(Float64(Float64(-1.0 / eps) - -1.0) * fma(-1.0, x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-299], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.7e+26], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.9e-246

   1. Initial program 70.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x < 9.5000000000000001e-299

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5 \]

   if 9.5000000000000001e-299 < x < 5.7000000000000003e26

   1. Initial program 62.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 68.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + 1}\right), x, 2\right) \cdot 0.5 \]

   if 5.7000000000000003e26 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 22.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(x \cdot \color{blue}{\left(\varepsilon - 1\right)}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.9%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot \color{blue}{x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(x, \varepsilon, x\right), 1\right)}{2} \]

   12. Taylor expanded in eps around 0

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \color{blue}{-1 \cdot x}\right)}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 39.8%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, \color{blue}{x}, 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \varepsilon - 1\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* eps eps) 1.0)))
   (if (<= x -2.9e-246)
     (* (fma (fma -1.0 (/ t_0 (- eps 1.0)) eps) x 2.0) 0.5)
     (if (<= x 9.5e-299)
       (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
       (* (fma (fma -1.0 (/ t_0 -1.0) (+ -1.0 eps)) x 2.0) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = (eps * eps) - 1.0;
	double tmp;
	if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (t_0 / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 9.5e-299) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else {
		tmp = fma(fma(-1.0, (t_0 / -1.0), (-1.0 + eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * eps) - 1.0)
	tmp = 0.0
	if (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 9.5e-299)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	else
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / -1.0), Float64(-1.0 + eps)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(t$95$0 / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 9.5e-299], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-1.0 * N[(t$95$0 / -1.0), $MachinePrecision] + N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-246

   1. Initial program 70.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x < 9.5000000000000001e-299

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5 \]

   if 9.5000000000000001e-299 < x

   1. Initial program 80.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 38.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{-1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{-1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.9e-246)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
   (* (- (fma (- x) (- 1.0 eps) 1.0) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.9e-246) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else {
		tmp = (fma(-x, (1.0 - eps), 1.0) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.9e-246)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(-x), Float64(1.0 - eps), 1.0) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.9e-246], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[((-x) * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e-246

   1. Initial program 70.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5 \]

   if -2.9e-246 < x

   1. Initial program 77.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.8%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right) - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.0%

       \[\leadsto \left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-271)
   (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
   (* (- (fma (- x) (- 1.0 eps) 1.0) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-271) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else {
		tmp = (fma(-x, (1.0 - eps), 1.0) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-271)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(-x), Float64(1.0 - eps), 1.0) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-271], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[((-x) * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999985e-271

   1. Initial program 69.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5 \]

   if -3.99999999999999985e-271 < x

   1. Initial program 77.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.2%

      \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right) - -1\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.5%

       \[\leadsto \left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (- (fma (- x) (- 1.0 eps) 1.0) -1.0) 0.5))

C

double code(double x, double eps) {
	return (fma(-x, (1.0 - eps), 1.0) - -1.0) * 0.5;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(-x), Float64(1.0 - eps), 1.0) - -1.0) * 0.5)
end

Wolfram

code[x_, eps_] := N[(N[(N[((-x) * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 74.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
4. Step-by-step derivation

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 62.4%

   \[\leadsto \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - -1\right) \cdot 0.5 \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(1 + -1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)\right) - -1\right) \cdot \frac{1}{2} \]
10. Step-by-step derivation

11. Applied rewrites 47.1%

    \[\leadsto \left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5 \]
12. Add Preprocessing

Alternative 17: 44.5% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 74.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 41.9%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))